Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 154 x^{2} + 6241 x^{4}$ |
| Frobenius angles: | $\pm0.0358886479470$, $\pm0.964111352053$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{78})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6088$ | $37063744$ | $243086686600$ | $1516233883567104$ | $9468276079987089928$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $80$ | $5934$ | $493040$ | $38927614$ | $3077056400$ | $243085917678$ | $19203908986160$ | $1517108713301374$ | $119851595982618320$ | $9468276077347332654$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=30 x^6+33 x^5+48 x^4+37 x^3+53 x^2+21 x+51$
- $y^2=48 x^6+78 x^5+55 x^4+9 x^3+39 x^2+57 x+53$
- $y^2=10 x^6+69 x^5+56 x^4+57 x^2+9 x+67$
- $y^2=76 x^6+9 x^5+52 x^4+15 x^3+53 x^2+9 x+38$
- $y^2=58 x^6+17 x^5+48 x^4+12 x^2+73 x+33$
- $y^2=38 x^6+33 x^5+76 x^4+49 x^3+32 x^2+13 x+33$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79^{2}}$.
Endomorphism algebra over $\F_{79}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{78})\). |
| The base change of $A$ to $\F_{79^{2}}$ is 1.6241.afy 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-78}) \)$)$ |
Base change
This is a primitive isogeny class.